\documentclass[11 pt]{article} \usepackage{amsmath,amsthm,amsfonts,amssymb,titlesec} \usepackage{hyperref} \usepackage{tikz} \usepackage{verbatim} \usepackage{accents} \usepackage[citestyle=alphabetic,bibstyle=alphabetic,backend=bibtex]{biblatex} \usepackage{todonotes} \usepackage[american]{babel} \usepackage{fancyhdr} \hypersetup{colorlinks=false} \usetikzlibrary{calc, decorations.pathreplacing,shapes.misc} \usetikzlibrary{decorations.pathmorphing} \usepackage[left=1in,top=1in,right=1in]{geometry} \usepackage[capitalize]{cleveref} \newcommand{\mathcolorbox}[2]{\colorbox{#1}{$\displaystyle #2$}} \newcommand{\xxx}{T base with combinatorial potential data } \newcommand{\Xxx}{T base with combinatorial potential data } \newcommand{\xxxc}{combinatorial potential stratified space } \newcommand{\Xxxc}{combinatorial potential stratified space } \newcommand{\argument}{symplectic character } \newcommand{\arguments}{symplectic characters } \newcommand{\snip}[2]{#1} 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\DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\TropB}{TropB} \DeclareMathOperator{\weight}{wt} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Fuk}{Fuk} \DeclareMathOperator{\str}{star} \DeclareMathOperator{\Ob}{Ob} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\CritVal}{CritV} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\FS}{FS} \DeclareMathOperator{\Vect}{Vect} \DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\Bl}{Bl} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Tw}{Tw} \DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\Arg}{\mathbf{M}}\begin{filecontents}{references.bib} @article{ballard2012hochschild, title={Hochschild dimensions of tilting objects}, author={Ballard, Matthew and Favero, David}, journal={International Mathematics Research Notices}, volume={2012}, number={11}, pages={2607--2645}, 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In mirror symmetry, we have pairs of Calabi-Yau manifolds $X, Y$ whose symplectic and algebraic invariants are equivalent. A precise version of this statement is the homological mirror symmetry conjecture: \begin{figure} \label{fig:HmsDiagram} \centering \begin{tikzpicture} \fill[gray!20,rounded corners] (3.5,1) rectangle (-6,2); \fill[gray!20,rounded corners] (3.5,-1) rectangle (-6,0); \node (v3) at (-2,1.5) {$D\text{Fuk}(X)$}; \node (v2) at (2,1.5) {$D\text{Fuk}(Y)$}; \node (v1) at (-2,-0.5) {$D\text{Coh}(X)$}; \node (v4) at (2,-0.5) {$D\text{Coh}(Y)$}; \draw[<->] (v1) edge (v2); \node at (-2,3) {$X$}; \node at (2,3) {$Y$}; \draw[dashed] (0,3.5) -- (0,-1.5); \draw[<->] (v3) edge (v4); \node at (-5,1.5) {$A$-model}; \node at (-5,-0.5) {$B$-model}; \end{tikzpicture} \caption{Mirror symmetry exchanges the symplectic invariants (\(A\)-side) and complex invariants (\(B\)-side) on a pair of ``mirror'' spaces.} \end{figure} When $X$ is \emph{Fano}, the mirror $Y = (Y, W)$ is a Landau-Ginzburg model, where \begin{itemize} \item $Y$ is an algebraic variety over $\CC$ \item $W: Y \to \CC$ is a holomorphic function satisfying additional conditions that we will discuss later. \end{itemize} In this case, our equivalence of invariants needs to be slightly modified: \begin{figure} \label{fig:HmsFanoDiagram} \centering \begin{tikzpicture} \fill[gray!20,rounded corners] (3.5,1) rectangle (-6,2); \fill[gray!20,rounded corners] (3.5,-1) rectangle (-6,0); \node (v3) at (-2,1.5) {$D\text{Fuk}(X)$}; \node (v2) at (2,1.5) {$D\text{FS}(Y,W)$}; \node (v1) at (-2,-0.5) {$D\text{Coh}(X)$}; \node (v4) at (2,-0.5) {$D\text{Sing}(Y,W)$}; \draw[<->] (v1) edge (v2); \node at (-2,3) {$X$}; \node at (2,3) {$(Y,W)$}; \draw[dashed] (0,3.5) -- (0,-1.5); \draw[<->] (v3) edge (v4); \node at (-5,1.5) {$A$-model}; \node at (-5,-0.5) {$B$-model}; \end{tikzpicture} \caption{Mirror symmetry exchanges Fano varieties with Landau-Ginzburg models.} \end{figure} Where $\FS(Y, W)$ is the Fukaya-Seidel category, and $\Sing(Y, W)$ is the category of singularities. In the literature, there is more work in the direction of $D\Fuk(X) \leftrightarrow D\Sing(Y, W)$, but in this reading group, we will focus on $D\Coh(X)\leftrightarrow D\Sing(Y, W)$. We call $D\Coh(X)$ the \emph{B}-side, while $D\FS(Y, W)$ is the \emph{A}-side. \section{B-side invariants} \label{art:BSideInvariants} \begin{definition} \label{def:DgCategoryOfCoherentSheaves} Let $X$ be an algebraic variety. Denote by $D\Coh(X)$ the dg-enhancement of the bounded derived category of coherent sheaves on $X$. \end{definition} \begin{definition} \label{def:SemiOrthogonalDecomposition} Let $\mathcal{C}$ be a triangulated category. A \emph{semi-orthogonal decomposition} of $\mathcal{C}$ is the data \[\mathcal{C} = \langle \mathcal{C}_0, \ldots, \mathcal{C}_i\rangle\] where \begin{itemize} \item $\mathcal{C}_i$ are full triangulated subcategories \item $\hom(\mathcal{C}_i, \mathcal{C}_j) = 0$ whenever $C_i \in \mathcal{C}_i, C_j \in \mathcal{C}_j$ and $j > i$ \item The $\mathcal{C}_i$ generate the category. \end{itemize} When $\mathcal{C}_i = D\Coh(\bullet) = D\Vect$, we say that $\mathcal{C}$ admits a full exceptional collection. \end{definition} When $X$ is Fano, typically $D\Coh(X)$ admits a semi-orthogonal decomposition \cref{def:SemiOrthogonalDecomposition} \[D\Coh(X) = \langle \mathcal{C}_1, \cdots \mathcal{C}_n\rangle\] \begin{example} \label{exm:BeilinsonCollection} When $X = \mathbb{P}^n$ or a toric Fano variety, then $D^b\Coh(X)$ admits a full exceptional collection. \end{example} \begin{example} \label{exm:AVarietyWithNoFullExceptionalCollection} It's easy to find varieties without a full exceptional collection. Suppose that $X$ admits a full exceptional collection. Then $H^{p, q}(X) = 0$ for $p \neq q$. \end{example} When we don't have a full exceptional collection, we can try to build one as best as we can and look at what is left. Typically, we have a semi-orthogonal decomposition \[D\Coh(X) = \langle \text{Ku}(X), \mathcal{C}_1, \ldots, \mathcal{C}_n\rangle\] where $\mathcal{C}_i = D\Coh(\bullet)$, and a larger part $\text{Ku}(X)$, which is often called the \emph{Kuznetsov component} of $D\Coh(X)$. Often, this Kuznetsov component contains information about the birational geometry of $X$. It can also reveal hidden relationships, e.g., it may be that $D\Coh(X) \neq D\Coh(X')$, but we have $\text{Ku}(X) = \text{Ku}(X')$. \section{\(A\)-side Landau-Ginzburg model} \label{art:ASideLandauGinzburgModel} On this side of the mirror, $Y$ will be a K\"ahler manifold, along with a holomorphic function $W: Y \to \CC$ which is: \begin{itemize} \item Proper (or controlled at $\infty$ in the fiber directions) \item A fibration near $\infty$ or has smooth fibers. \end{itemize} The objects of the Fukaya-Seidel category $\FS(Y, W)$ are Lagrangian submanifolds $L \subset Y$ such that $W|_L$ is a fibration over $\RR_+$ outside of a compact set. \begin{figure} \label{fig:LagrangiansInTheFukayaSeidelCategory} \centering \begin{tikzpicture} \fill[gray!20] (-3,2.5) rectangle (4,-1.5); \draw[fill=blue!20] (2.5,0.5) .. controls (2,0.5) and (1,1.5) .. (0.5,1.5) .. controls (0,1.5) and (-1.5,1) .. (-1.5,0.5) .. controls (-1.5,0) and (-1,-1) .. (-0.5,-1) .. controls (0,-1) and (0,-0.5) .. (0.5,-0.5) .. controls (1,-0.5) and (1,-1) .. (1.5,-1) .. controls (2,-1) and (2,0.5) .. (2.5,0.5) .. controls (4,0.5) and (3,0.5) .. (4,0.5); \node at (0.5,0.5) {$W(L)$ }; \node at (3.5,2) {$\mathbb{C}$}; \end{tikzpicture} \caption{A Lagrangian submanifold in a Fukaya-Seidel category needs to have prescribed behavior going off to infinity.} \end{figure} To take the Lagrangian intersection Floer cohomology between two such Lagrangians, we need to perturb the Lagrangians at infinity; the perturbation we take will rotate one Lagrangian at infinity. As a vector space, $\hom(L, K) = \CC(\langle L \cap \phi(K)\rangle)$. \begin{figure} \label{fig:PushoffInFukayaSeidelCategory} \centering \begin{tikzpicture} \fill[gray!20] (-3,2.5) rectangle (4,-1.5); \draw[fill=red!20] (2.5,0.5) .. controls (2,0.5) and (1,1.5) .. (0.5,1.5) .. controls (0,1.5) and (-1.5,1) .. (-1.5,0.5) .. controls (-1.5,0) and (-1,-1) .. (-0.5,-1) .. controls (0,-1) and (0,-0.5) .. (0.5,-0.5) .. controls (1,-0.5) and (1,-1) .. (1.5,-1) .. controls (2,-1) and (2,0.5) .. (2.5,0.5) .. controls (4,0.5) and (3,0.5) .. (4,0.5); \node at (0.5,1) {$W(L)$ }; \node at (3.5,2) {$\mathbb{C}$}; \draw[fill=blue, opacity=.5] (-0.5,0) .. controls (-1,0) and (-2,0) .. (-2,-1) .. controls (-2,-1.5) and (-0.5,-1.3) .. (0,-1.3) .. controls (0.5,-1.3) and (1.5,-0.5) .. (2,-0.5) .. controls (2.5,-0.5) and (3.5,1.5) .. (4,1.5) .. controls (3.5,1.5) and (2.5,-0.5) .. (2,-0.5) .. controls (1.5,-0.5) and (0,0) .. (-0.5,0); \node at (-0.5,-0.5) {$W(\phi(K))$}; \end{tikzpicture} \caption{To obtain transversality, the Lagrangian submanifolds in the Fukaya-Seidel category are pushed off one another with respect to the projection \(W: Y \to \CC\).} \end{figure}The differential and product structure on hom-spaces is given by counts of pseudo-holomorphic disks. \subsection{Comparison to \(B\)-side} \label{art:ComparisonToBSide} Let $\CritVal(W) \subset \CC$ be the subset of critical values of $W$. For each $c \in \CritVal(W)$, we have a ``smaller'' Landau-Ginzburg model $Y_c$ by trivially extending $W^{-1}(B(c, \epsilon)) \to B(c, \epsilon)$ to a fibration over $\CC$. \begin{figure} \label{fig:SmallNeighborhoodFsCategory} \centering \begin{tikzpicture} \draw (-3,2.5) rectangle (4,-1.5); \node at (3.5,2) {$\mathbb{C}$}; \node at (-1,0) {$\times$}; \node at (1.5,0.5) {$\times$}; \node at (-1,1.5) {$\times$}; \draw[dotted] (-1,1.5) ellipse (0.5 and 0.5); \node at (0,1.5) {$B(c, \epsilon)$}; \end{tikzpicture} \caption{A small neighborhood in the base of a symplectic LG model can be used to build another LG model.} \end{figure}Define $\FS_c = \FS(Y_c, W)$. \begin{definition} \label{def:VanishingPath} Let $c \in \CritVal(W)$ be a critical point. A \emph{vanishing path} for $c$ is a path $\gamma: [0, \infty) \to \CC$ such that \begin{itemize} \item $\gamma(0) = c$ \item $\gamma$ avoids all other critical values \item $\gamma(t) = t$ for $t \gg 0$. \end{itemize} \end{definition} A vanishing path determines an embedding $\FS_c \to \FS(Y, W)$ by extending Lagrangian submanifolds over the vanishing path using symplectic parallel transport. \begin{figure} \label{fig:ExtendingByVanishingPath} \centering \begin{tikzpicture} \fill[gray!20] (-3,2.5) rectangle (4,-1.5); \node at (3.5,2) {$\mathbb{C}$}; \node at (-1,0) {$\times$}; \node at (1.5,0.5) {$\times$}; \node at (-1,1.5) {$\times$}; \draw[dotted] (-1,1.5) ellipse (0.5 and 0.5); \begin{scope}[] \draw[dotted, fill=gray!20] (-7,1.5) ellipse (1 and 1); \draw[fill=blue!20] (-6,1.5) .. controls (-6.1,1.5) and (-6.4,1.5) .. (-6.5,1.5) .. controls (-6.6,1.5) and (-6.9,1.8) .. (-7.1,1.8) .. controls (-7.3,1.8) and (-7.4,1.5) .. (-7.4,1.4) .. controls (-7.4,1.2) and (-7.1,1.2) .. (-6.9,1.2) .. controls (-6.7,1.2) and (-6.6,1.5) .. (-6.5,1.5); \end{scope} \begin{scope}[scale=0.5, shift={(5,1.5)}] \draw[dotted] (-7,1.5) ellipse (1 and 1); \draw[fill=blue!20] (-6,1.5) .. controls (-6.1,1.5) and (-6.4,1.5) .. (-6.5,1.5) .. controls (-6.6,1.5) and (-6.9,1.8) .. (-7.1,1.8) .. controls (-7.3,1.8) and (-7.4,1.5) .. (-7.4,1.4) .. controls (-7.4,1.2) and (-7.1,1.2) .. (-6.9,1.2) .. controls (-6.7,1.2) and (-6.6,1.5) .. (-6.5,1.5); \end{scope} \draw (-0.5,1.5) .. controls (1,1.5) and (2,0.5) .. (2.5,0.5) .. controls (3,0.5) and (3.5,0.5) .. (4,0.5); \node at (3,0) {$\gamma$}; \end{tikzpicture} \caption{A vanishing path determines a method for extending Lagrangians belonging to the Fukaya-Seidel category near a critical value to the Fukaya-Seidel category of \((Y, W)\).} \end{figure}If we choose ``disjoint'' vanishing paths, we get a semi-orthogonal decomposition: \[\FS(Y, W) = \langle \FS_1, \ldots, \FS_n\rangle \] \begin{example} \label{exm:MirrorSymmetryForTheProjectivePlane} \label{exm:HMSforP2} Consider the symplectic Landau-Ginzburg model whose critical points are arranged as follows: \begin{figure} \label{fig:VanishingPathsForP2} \centering \begin{tikzpicture} \fill[gray!20] (-3,2.5) rectangle (4,-1.5); \node at (3.5,2) {$\mathbb{C}$}; \node[green] at (-1,0) {$\times$}; \node[blue] at (1.5,0.5) {$\times$}; \node[red] at (-1,1.5) {$\times$}; \node[below,green] at (-1,0) {$1$}; \node[below,blue] at (1.5,0.5) {$2$}; \node[below,red] at (-1,1.5) {$3$}; \draw[red] (-1,1.5) .. controls (0.5,1.5) and (2,0.5) .. (2.5,0.5) .. controls (3,0.5) and (3.5,0.5) .. (4,0.5); \draw[blue] (1.5,0.5) -- (4,0.5); \draw[green] (-1,0) .. controls (-1.5,-0.5) and (-2,1) .. (-2,1.5) .. controls (-2,2) and (-0.5,2) .. (0,2) .. controls (0.5,2) and (3,0.5) .. (4,0.5); \end{tikzpicture} \caption{A set of vanishing paths for the Lefschetz fibration \(W(z_1, z_2) = z_1 + z_2 + (z_1z_2)^{-1}\)} \end{figure} We obtain a semi-orthogonal decomposition $\FS(Y, W) = \langle \FS_1, \FS_2, \FS_3 \rangle$. \end{example} \subsection{Exceptional collections} \label{art:ExceptionalCollections} Suppose additionally that $W^{-1}(c)$ has an $A_1$ singularity (a node) then \[\FS_c = D\Coh(\text{pt}) = D\Vect\] and the generating object is called the \emph{vanishing thimble} associated with the point. So if all critical points of $W$ are more (i.e., $W$ is a Lefschetz fibration), then $\FS(Y, W)$ admits a full exceptional collection. If $W$ has isolated singularities, they can be ``Morsified'' by a small perturbation (in the symplectic category). From this, we obtain the following principle: if $(Y, W)$ is a symplectic Landau-Ginzburg model and $W$ has isolated singularities, then $\FS(Y, W)$ admits a full exceptional collection. This tells us that many examples arising from mirror symmetry will \emph{not} have isolated singularities, as their mirror spaces will not have full exceptional collections! However, we still have the following expectations. When $X$ is Fano, and $(Y, W)$ is its mirror, then we expect $W$ will have some isolated singularities, and an additional critical value $c_{\text{Ku}}$ with non-isolated singularities with \[\text{Ku}(X) = \FS_{c_{\text{Ku}}}.\] \printbibliography \end{document}